Assignment 4 | Complete Solution

Textbook problems:
3.14    Calculate the time necessary to perform a multiply using the approach given in Figures 3.4 and 3.5 if an integer is 8 bits wide and each step of the operation takes 4 time units. Assume that in step 1a an additional is always performed – either the multiplicand will be added, or a zero will be. Also assume that the registers have already been initialized (you are just counting how long it take sot do the multiplication loop itself). If this is being done in hardware, the shifts of the multiplicand and multiplier can be done simultaneously. If this is being done in software, they will have to be done one after the other. Solve for each case.
 
3.15    Calculate the time necessary to perform a multiply using the approach described in the test (32 adders stacked vertically) if an integer is 8 bits wide and an adder takes 4 time units.
3.16    Calculate the time necessary to perform a multiply using the approach given in Figure 3.7 if an integer is 8 bits wide and an adder takes 4 time units.
Additional Problems
1.    Use Boolean algebra to find the minimum cost SOP expression for the function below.
(𝑎, 𝑏, 𝑐, 𝑑) = 𝑎𝑏̅𝑐 + 𝑐𝑑 + 𝑎̅𝑑 + 𝑏𝑑 2.

2.    Use Boolean algebra to find the minimum cost POS expression for the function below.
(𝑎, 𝑏, 𝑐, 𝑑, 𝑒) = ∏𝑀(0,8,9,13,14)

3.    Answer the following questions using the circuit in Figure 1.
 
a.    Derive the truth table for the circuit in Figure 1.

b.    Derive the Boolean expression from the truth table (not simplified).

c.    Redesign the circuit so that it only uses two-input NAND gates.

4.    Show how to implement the function below using a 3:8 decoder and an OR gate.
(𝑎, 𝑏, 𝑐) = ∑𝑚(0,2,3,4,5,7)

5.    Assume that the integers are unsigned. You must perform the operation in binary and show your work. Use ten bits of precision and state whether there was an overflow – justify your answer.
     188
+   371

6.    Assume that the integers are signed, 2’s compliment. You must perform the operation in binary and show your work. Use ten bits of precision and state whether there was an overflow – justify your answer
    188
-   371

7.    Assume that the integers are unsigned. You must perform the operation in binary and show your work. Use sixteen bits of precision.
    188
X    64

8.    Assume that the integers are unsigned. You must use hexadecimal arithmetic. Determine the result in hex, octal, and binary.
    6A  94
+  5AB1

9.    Show how to divide 371 by 8 using unsigned binary division. Show the remainder, if there is one.

10.    Show how to perform the operation (6 * 9) using the optimized multiplier discussed in class. Assume the operands are 5b instead of 32b (be sure that you adjust all of the registers appropriately). You must show each step, from initialization to the final product.

11.    Show how to perform the operation (-6 * 9) using Booth’s algorithm. Assume the operands are 5b instead of 32b (be sure that you adjust all of the registers appropriately). You must show each step, from initialization to the final product.